On small sets of integers

An upper quasi-density on $\bf H$ (the integers or the non-negative integers) is a real-valued subadditive function $\mu^\ast$ defined whole power set of $\mathbf such that $\mu^\ast(X) \le \mu^\ast({\bf H}) = 1$ and $\mu^\ast(k \cdot X + h) \frac{1}{k}\, \mu^\ast(X)$ for all $X \subseteq \bf H$, $k \in {\bf N}^+$, $h N$, where := \{kx: x X\}$; an density non-decreasing with respect to inclusion. We say small if 0$ every H$. Main examples densities are given by analytic, Banach, Buck, P\'olya densities, along uncountable family $\alpha$-densities, $\alpha$ real parameter $\ge -1$ (most notably, $\alpha corresponds logarithmic density, asymptotic density). It turns out subset only it belongs zero Buck Z$. This allows us show many interesting sets small, including less than fixed number prime factors, counted multiplicity; numbers represented binary quadratic form integer coefficients whose discriminant not perfect square; image Z$ through non-linear integral polynomial in one variable.